$p$-adic family 2.1.8.28b1.7-1.2.4a

• Label: 2.1.8.28b1.7-1.2.4a
• Base: 2.1.8.28b1.7
• Degree: \(2\)
• e: \(2\)
• f: \(1\)
• c: \(4\)

Defining polynomial

$x^{2} + \left(b_{5} \pi^{3} + a_{3} \pi^{2}\right) x + c_{6} \pi^{4} + \pi$

Invariants

Residue field characteristic:	$2$
Degree:	$2$
Base field:	2.1.8.28b1.7
Ramification index $e$:	$2$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$4$
Absolute Artin slopes:	$[3,\frac{7}{2},4,\frac{17}{4}]$
Swan slopes:	$[3]$
Means:	$\langle\frac{3}{2}\rangle$
Rams:	$(3)$
Field count:	$4$ (complete)
Ambiguity:	$2$
Mass:	$2$
Absolute Mass:	$1$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group:	$C_2^4:D_4$
Hidden Artin slopes:	$[2,2]^{2}$
Indices of inseparability:	$[45,38,28,16,0]$
Associated inertia:	$[1,1,1,1]$
Jump Set:	$[1,3,7,15,31]$

Fields

Showing all 4

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Label	Polynomial $/ \Q_p$	Galois group $/ \Q_p$	Galois degree $/ \Q_p$	$\#\Aut(K/\Q_p)$	Artin slope content $/ \Q_p$	Swan slope content $/ \Q_p$	Hidden Artin slopes $/ \Q_p$	Hidden Swan slopes $/ \Q_p$	Ind. of Insep. $/ \Q_p$	Assoc. Inertia $/ \Q_p$	Resid. Poly	Jump Set
2.1.16.60o1.173	$x^{16} + 8 x^{13} + 4 x^{12} + 8 x^{10} + 4 x^{8} + 8 x^{6} + 2$	$C_2^4:D_4$ (as 16T392)	$128$	$4$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,2]^{2}$	$[1,1]^{2}$	$[45, 38, 28, 16, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.60o1.205	$x^{16} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{10} + 12 x^{8} + 8 x^{6} + 16 x^{4} + 16 x + 2$	$C_2^4:D_4$ (as 16T392)	$128$	$4$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,2]^{2}$	$[1,1]^{2}$	$[45, 38, 28, 16, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.60o1.220	$x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 4 x^{8} + 8 x^{6} + 24 x^{4} + 16 x + 2$	$C_2^4:D_4$ (as 16T392)	$128$	$4$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,2]^{2}$	$[1,1]^{2}$	$[45, 38, 28, 16, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$
2.1.16.60o1.228	$x^{16} + 8 x^{15} + 8 x^{13} + 4 x^{12} + 12 x^{8} + 8 x^{6} + 8 x^{4} + 2$	$C_2^4:D_4$ (as 16T392)	$128$	$4$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,2]^{2}$	$[1,1]^{2}$	$[45, 38, 28, 16, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 3, 7, 15, 31]$

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